14:00h – 14:40h

A result of multiplicity of nontrivial solutions to an elliptic Kirchhoff-Boussinesq equation type in $\mathbb{R}^{N}$

In this work, using  Ljusternik-Schnirelman category theory, we study the existence and multiplicity of nontrivial solutions  for the following class of elliptic Kirchhoff-Boussinesq type problems given by

$$\varepsilon^4 \Delta^2 u \pm \varepsilon^p \Delta_p u + V(x)u=f(u) + \beta|u|^{2_{**}-2}u ~~\text{in}~~\mathbb{R}^N,\\ u \in H^2(\mathbb{R}^N),$$

where $\varepsilon>0$, $2<p<2^{*}=\frac{2N}{N-2}$,  $2_{**}=\frac{2N}{N-4}$ and $N\geq5$, $V:\mathbb{R}^N \to\mathbb{R}$ is a continuous function and $f:\mathbb{R} \to \mathbb{R}$ is a function of $C^{1}$ class. We consider the subcritical case, i.e, $\beta=0$ and critical case, i.e, $\beta=1$.

14:50h – 15:30h

Roughness-induced effects on a reaction-diffusion equation defined in a thin domain

 In this talk we discuss a reaction-diffusion problem in a thin plane region, endowed with a Robin-type boundary condition, which describes reactions catalyzed at the boundary. Motivated by microfluidic applications we allow, in principle, resonant rough behavior in which the amplitude and period of the roughness at the boundary have the same scale as the domain thickness.

   14:00h – 14:40h

Integration with values in a Neumann space for PDE’s

We It may be useful to use elements in spaces such as L^{p}([0,T] ;L^{q}_{loc}(Ω)) or H^{-1}([0,T]; W^{m,q}(Ω)-weak) … unfortunately they were not yet defined. Theses spaces L^{q}_{loc}(Ω) and W^{m,q}(Ω)-weak are Neumann spaces, that are vector spaces provided with semi-norms that made them separated and sequentially complete.

We will define D'(Ω;E), L^{p}(Ω;E) and W^{m,p}(Ω;E) where E is a Neumann space and Ω is an open set of R^{d}, and we show some applications to PDE’s.
We insist on L^{p}(Ω;E) which is composed of measures: it cannot be composed
of class of almost equal functions because the theory fails, since Egoroff theorem
does not extend if E is not metrisable.
When E is a Banach space and \hat{f} \hat{L}^{p}(Ω;E), that is the space of p-integrable
almost equal functions, then a measure \bar{f} is defined by: for all φ \in K(Ω),
<\bar{f}, φ>=\int_{Ω}\hat{f}φ
We prove that our space L^{p}(Ω;E) is {\bar{f} : \hat{f} \in \hat{L}^{p}(Ω;E)} and that \hat{f}→\bar{f} is an isomorphism. So, our L^{p}(Ω;E) is the true Lebesgue space if E is a Banach space.

14:50h – 15:30h

Decay estimates for semilinear wave equations with time-dependent time delay

14:00h – 14:40h

Unique Continuation Principle for  Nonlocal  Nonlinear Dispersive Models.

14:50h – 15:30h

Optimal control of PDEs: Some environmental applications

 14:00h – 14:40h

Superlinear Convergence of a Semismooth Newton Method for some Optimization Problems with Applications to Control Theory

Let (X, S, μ) be a measure space with μ(X) < ∞. In this talk, we prove the superlinear convergence of a semismooth Newton method to solve the following abstract optimization problem:
(P)     min J(u) +k/2(//u//^2L2(X))
α≤u(x)≤β a.e.[μ]

,where κ > 0, −∞ ≤ α < β ≤ +∞, and J : Lp(X) → R is a function of

class C2 for some p ∈ [2, +∞). Many optimal control problems fall within this abstract formulation, such as distributed or boundary control problems and bilinear control problems associated with nonlinear elliptic or parabolic equations. We propose a superlinearly convergent semismooth Newton method to compute a local minimizer ̄u of (P) assuming that the no-gap second order sufficient optimality condition and the strict complementarity condition are fulfilled.

These assumptions are usually imposed to prove the superlinear or second order convergence of numerical algorithms for solving finite dimensional optimization problems. The translation of our algorithm to the case of optimal control problems governed by elliptic or parabolic semilinear equations is immediate.

14:50h – 15:30h

Least-Squares Pressure Recovery in Reduced Order Methods for Incompressible Flows